The vertices of a triangle are labeled clockwise A(–5, 3), B(6, 1), and C(–2, –3). How could you show that the figure is a right
triangle?
1 answer:
Explanation:
When the points are plotted on a graph, it is easy to see that the slope of AC is -2 and the slope of BC is 1/2. These slope values have a product of -1, so the corresponding line segments are perpendicular to each other.
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If you have studied vectors, you can find the dot product of AC with BC:
AC = (-5, 3) -(-2, -3) = (-3, 6)
BC = (6, 1) -(-2, -3) = (8, 4)
The dot product is ...
(-3, 6)·(8, 4) = (-3)(8) + (6)(4) = -24+24 = 0
When the dot product of vectors is zero, they are perpendicular.
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Step 1: Writing Null and Alternate Hypothesis
First we need to write the Null and Alternate Hypothesis for this problem.
Researcher claims that mileage will be increased on addition of Fuel additive. So the null and alternate hypothesis will be:
: μ ≤ 25.1 (Null Hypothesis)
: μ > 25.1 (Alternate Hypothesis)
This is a Right Tailed Test. Since population standard deviation is not know, we will use t-test to check the researchers claim.
Step 2: Finding Test Statistic
Sample Mean = x = 26.8
Standard Deviation = s = 3.9
Sample Size = n = 35
Degrees of Freedom = df = n - 1 = 34
Test statistic(t) is given by:
Step 3: Finding p value
Using t table or calculators find the p value for t=2.579 with 34 degrees of freedom for one tailed test.
P value comes out to be:
p = 0.0072
Step 4: Conclusion
Since the p value is lesser than the significance level of 0.05, we reject the Null Hypothesis.
We have enough evidence to support the researcher's claim that the new additive increases the miles per gallon of the cars.
First we need to write the Null and Alternate Hypothesis for this problem.
Researcher claims that mileage will be increased on addition of Fuel additive. So the null and alternate hypothesis will be:
This is a Right Tailed Test. Since population standard deviation is not know, we will use t-test to check the researchers claim.
Step 2: Finding Test Statistic
Sample Mean = x = 26.8
Standard Deviation = s = 3.9
Sample Size = n = 35
Degrees of Freedom = df = n - 1 = 34
Test statistic(t) is given by:
Step 3: Finding p value
Using t table or calculators find the p value for t=2.579 with 34 degrees of freedom for one tailed test.
P value comes out to be:
p = 0.0072
Step 4: Conclusion
Since the p value is lesser than the significance level of 0.05, we reject the Null Hypothesis.
We have enough evidence to support the researcher's claim that the new additive increases the miles per gallon of the cars.
Slope-intercept form:
y = mx + b
"m" is the slope, "b" is the y-intercept (the y value when x = 0)
You need to find "m" and "b"
1.) For lines to be perpendicular, their slopes have to be the opposite/negative reciprocal (flipped sign and number)
For example:
slope is 2
perpendicular line's slope is -1/2
slope is -2/3
perpendicular line's slope is 3/2
The given line's slope is -4, so the perpendicular line's slope is 1/4.
y = 1/4x + b
To find "b", you can plug in the point (4, -2) into the equation
y = 1/4x + b
-2 = 1/4(4) + b
-2 = 1 + b
-3 = b
2.) To find "m", use the slope formula and plug in the two points:
y = -3/2x + b
Plug in one of the points to find "b"
y = -3/2x + b
-1 = -3/2(4) + b
-1 = -6 + b
5 = b
3.) y-intercept: 5 (since the given equation's y-intercept is 5)
I'm confused with what they mean by "steep", so I'll try to update this later unless someone else has an answer
So I looked it up, and it says that a steep slope is a line that is more vertical [if that makes sense]. So you could do a slope of +3 (more than 3, like 4, 5, 6...) because the given line's slope is 3, and you need a new line that is more steeper(vertical)